If you can’t send a selfie from a rock concert or an NFL game, does it still count as being there? That’s more than an existential question as texting, tweeting, Snapchatting and Instagramming fans outpace—and overload—networks at public events.

At Lollapalooza this summer, four young women huddled amid the crowd, struggling with their phones. The cellular networks serving Chicago’s Grant Park, the site of the annual music festival,...